On odd harmonious labelling of even cycles with parallel chords and dragons with parallel chords

Abstract

Labelling in graph theory is an active area of research due to its wide range of applications. A graph labelling is an assignment of integers to the vertices (or) edges (or) both subject to certain conditions. This paper deals with one such labelling called odd harmonious labelling. A graph G = (V, E) with |V (G)| = p and |E (G)| = q is said to be odd harmonious if there exist an injection f : V (G) → {0, 1, 2, …, 2q – 1} such that the induced function f* : E (G) → {1, 3, 5, …, 2q – 1} defined by f*(uv) = f(u) + f(v) is bijective. In this paper we prove that every even cycle Cn (n ≥ 6) with parallel P3 chords is odd harmonious. We also prove that the disjoint union of two copies of even cycle Cn with parallel P3 chords and the joint sum of two copies of even cycle Cn with parallel P3 chords is odd harmonious. Moreover we show that the chain of even cycles Cn (n ≥ 6) with parallel P3 chords, joining two copies of even cycles Cn by a path and also dragons with parallel chords obtained from every odd cycle Cn (n ≥ 7) after removing two edges from the cycle Cn, dragons with parallel P4 chords obtained from every odd cycle Cn (n ≥ 9) after removing two edges from the cycle Cn are odd harmonious.